Problem: $\sum\limits_{k=1}^{420 }{{(10k -54)}}=$
Explanation: What is the question asking for? The question is asking for the sum of the values of $10k -54$ from $k = 1$ to $k = 420 $ : $(10 \cdot 1 -54) + (10 \cdot 2 -54) +... + (10\cdot {420} -54)$ The series is arithmetic because the formula $10k -54$ is a linear function of $k$. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The number of terms $(n = {420})$ is the upper limit of the sigma notation. We need to find $a_1$ (the first term) and $a_{420}$ (the last term). Step 1: Find $a_1$ and $a_{420}$ (the first and the last term) $a_1 = 10(1) -54 = {-44}$ $a_{420} = 10(420) -54 = {4146}$ Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{420}}&= \dfrac {\left({-44} + {4146} \right)}{2} \cdot {420} \\\\ S_{{420}} &= 2051 \left(420\right) \\\\ S_{{420}} &= 861{,}420\end{aligned}$ The answer $ 861{,}420 $